Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.
Conceptual mathematics: a first introduction to categories
Conceptual mathematics by F. W. Lawvere, 2009, Cambridge University Press edition, in English - 2nd ed.
Infinity Category Theory Offers a Bird's-Eye View of Mathematics
Mathematicians have expanded category theory into infinite dimensions, revealing new connections among mathematical concepts
F. William Lawvere, Stephen H. Schanuel Conceptual Mathematics A First Introduction To Categories ( 2009, Cambridge University Press) ( 1)
Category Theory first introduction
Basic Category Theory
This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.
Mathematical Beauty, Truth and Proof in the Age of AI | Quanta Magazine
Mathematicians have started to prepare for a profound shift in what it means to do mathematics.
Kill Math
The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
Emily Riehl, A New Paradigm for Mathematical Proof? | Natural Philosophy Symposium 2025
Mathematics in type theory.
An explanation of how to set up mathematics using universes, types, and terms
Introduction To Category Theory
Introduction To Category Theory by Steve Awodey, 2010, Oxford University Press, USA, Oxford University Press edition,
Categorical logic and type theory
Categorical logic and type theory by Bart Jacobs, 1999, Elsevier Science edition, in English - 1st ed.
Mathematicians Discover the Perfect Way to Multiply | Quanta Magazine
By chopping up large numbers into smaller ones, researchers have rewritten a fundamental mathematical speed limit.
Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics)
Introduction to higher order catagorical logic by J. Lambek, March 25, 1988, Cambridge University Press edition, Paperback in English
Mathematics in the Library of Babel · Daniel Litt
Mathematics isn't only about saying true things. It's about asking the right questions, being confused, stumbling about, getting distracted, being wrong…
If math is more than proof, we need to better celebrate the rest of it
[This is a guest post by Grant Sanderson. This blog post was initially written in a different file format and converted using AI. — T.] A sentiment echoing throughout the mathematics communit…
General Theory of Natural Equivalences
Samuel Eilenberg, Saunders MacLane, General Theory of Natural Equivalences, Transactions of the American Mathematical Society, Vol. 58, No. 2 (Sep., 1945), pp. 231-294
The HoTTest Axiom of math